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is geometric, because each successive term can be obtained by multiplying the previous term by 1/.
Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a geometric series using only two terms, r and a. The term r is the common ratio, and a is the first term of the series. As an example the geometric series given in the introduction,
may simply be written as
The following table shows several geometric series with different common ratios:
|Common ratio, r||Start term, a||Example series|
|10||4||4 + 40 + 400 + 4000 + 40,000 + ···|
|1/3||9||9 + 3 + 1 + 1/3 + 1/9 + ···|
|1/10||7||7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···|
|1||3||3 + 3 + 3 + 3 + 3 + ···|
|−1/2||1||1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···|
|–1||3||3 − 3 + 3 − 3 + 3 − ···|
The behavior of the terms depends on the common ratio r:
The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely-many terms. The sum can be computed using the self-similarity of the series.
Consider the sum of the following geometric series:
This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on:
This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s from the original series s cancels every term in the original but the first:
A similar technique can be used to evaluate any self-similar expression.
For , the sum of the first n terms of a geometric series is:
where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:
As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes
When a = 1, this simplifies to:
the left-hand side being a geometric series with common ratio r. We can derive this formula:
The general formula follows if we multiply through by a.
The formula holds true for complex "r", with the same restrictions (modulus of "r" is strictly less than one).
Since (1 + r + r2 + ... + rn)(1−r) = 1−rn+1 and rn+1 → 0 for | r | < 1.
Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function:
So S converges to
For , the sum of the first n terms of a geometric series is:
We can derive this formula as follows:
A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:
The formula for the sum of a geometric series can be used to convert the decimal to a fraction:
The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:
Note that every series of repeating consecutive decimals can be conveniently simplified with the following:
That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part (as an integer) and 10n - 1.
Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.
Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.
Assuming that the blue triangle has area 1, the total area is an infinite sum:
The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives
This is a geometric series with common ratio 1/4 and the fractional part is equal to
The sum is
For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is
The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is
Thus the Koch snowflake has 8/5 of the area of the base triangle.
The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is taken to be half the remaining distance. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with .
For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + ), where is the yearly interest rate.
Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + )2 (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is
which is the infinite series:
This is a geometric series with common ratio 1 / (1 + ). The sum is the first term divided by (one minus the common ratio):
For example, if the yearly interest rate is 10% ( = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.
This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a security.
The formula for a geometric series
By differentiating the geometric series, one obtains the variant